cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A347762 Number of compositions (ordered partitions) of n into at most 2 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 7, 7, 6, 8, 7, 9, 7, 10, 8, 7, 5, 10, 7, 9, 9, 10, 7, 12, 7, 14, 10, 13, 8, 14, 5, 11, 8, 12, 7, 12, 7, 12, 10, 11, 5, 16, 7, 15, 8, 12, 5, 17, 6, 14, 8, 11, 5, 16, 7, 13, 8, 14, 6, 18, 5, 16, 10, 14
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2021

Keywords

Crossrefs

Cf. A000961, A280543, A282062, A347763, A347764.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Join[{1},Select[Range@n,PrimePowerQ]]],1],{n,0,70}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

A347763 Number of compositions (ordered partitions) of n into at most 3 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 15, 20, 24, 29, 34, 40, 45, 52, 55, 58, 62, 67, 69, 76, 79, 87, 88, 92, 89, 100, 96, 106, 109, 121, 111, 127, 125, 140, 139, 158, 149, 173, 152, 168, 159, 184, 156, 196, 168, 193, 173, 206, 173, 220, 186, 222, 201, 236, 185, 243, 203, 252
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2021

Keywords

Crossrefs

Cf. A000961, A280543, A282064, A347762, A347764.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,3,Join[{1},Select[Range@n,PrimePowerQ]]],1],{n,0,70}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

A347764 Number of compositions (ordered partitions) of n into at most 4 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 25, 40, 59, 81, 106, 136, 170, 208, 251, 294, 339, 383, 431, 476, 530, 583, 642, 696, 757, 804, 866, 914, 980, 1041, 1125, 1167, 1256, 1312, 1405, 1466, 1598, 1657, 1790, 1840, 1961, 2004, 2148, 2160, 2335, 2365, 2505, 2502, 2707, 2664, 2884
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2021

Keywords

Crossrefs

Cf. A000961, A280543, A341133, A347762, A347763.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,4,Join[{1},Select[Range@n,PrimePowerQ]]],1],{n,0,50}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

A331925 Number of compositions (ordered partitions) of n into distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 11, 17, 19, 48, 49, 62, 85, 120, 258, 175, 337, 464, 631, 646, 932, 1686, 1991, 2122, 2455, 4118, 4545, 6010, 6481, 13302, 14383, 16177, 16912, 26454, 32024, 35468, 42389, 57334, 107708, 73830, 125629, 142560, 200377, 172752, 244624
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			a(6) = 10 because we have [5, 1], [4, 2], [3, 2, 1], [3, 1, 2], [2, 4], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Links

Crossrefs

Cf. A000961, A023893, A106244, A219107, A280543, A331847.

Programs

  • Maple
    N:= 50: # for a(0)..a(N)
P:= select(isprime, [2,seq(i,i=3..N,2)]):
PP:= sort([1,seq(seq(p^j, j = 1 .. ilog[p](N)),p=P)]):G:= 1:
for s in PP do
  G:= G + series(G*x*y^s,y,N+1);
od:
G:= convert(G,polynom):
T:= add(coeff(G,x,i)*i!,i=0..N):
seq(coeff(T,y,i),i=0..N); # Robert Israel, Jun 28 2024

A347775 Number of compositions (ordered partitions) of n into at most 5 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 55, 94, 151, 227, 326, 450, 603, 786, 1005, 1259, 1543, 1856, 2201, 2571, 2978, 3417, 3896, 4402, 4950, 5506, 6104, 6710, 7366, 8040, 8792, 9526, 10342, 11150, 12042, 12918, 13977, 14975, 16145, 17242, 18514, 19628, 21015, 22170, 23671, 24940
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A000961, A280543, A341134, A347762, A347763, A347764, A347776.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Join[{1},Select[Range@n,PrimePowerQ]]],1],{n,0,46}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347776 Number of compositions (ordered partitions) of n into at most 6 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 115, 207, 353, 572, 882, 1305, 1863, 2581, 3491, 4615, 5974, 7583, 9462, 11616, 14070, 16844, 19965, 23436, 27289, 31502, 36104, 41074, 46462, 52244, 58534, 65230, 72458, 80118, 88352, 97011, 106448, 116393, 127189, 138532, 150819, 163473
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A000961, A280543, A341135, A347762, A347763, A347764, A347775.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,6,Join[{1},Select[Range@n,PrimePowerQ]]],1],{n,0,43}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A280605 Expansion of 1/(1 - Sum_{p prime, k>=2} x^(p^k)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0, 6, 5, 1, 0, 10, 10, 3, 0, 18, 23, 9, 2, 31, 46, 22, 6, 56, 94, 56, 19, 101, 184, 129, 50, 185, 364, 293, 134, 344, 708, 638, 332, 651, 1378, 1375, 805, 1265, 2665, 2901, 1878, 2503, 5161, 6057, 4306, 5061, 10005, 12488, 9653, 10384, 19461, 25556, 21319
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 06 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into proper prime powers (A246547).

Examples

			a(12) = 3 because we have [8, 4], [4, 8] and [4, 4, 4].

Links

Crossrefs

Cf. A246547, A280195, A280200, A280543, A280586.

Programs

  • Mathematica
    nmax = 67; CoefficientList[Series[1/(1 - Sum[Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
  • PARI
    x='x+O('x^68); Vec(1/(1 - sum(k=2, 67, sign(bigomega(k) - 1) * (1\omega(k)) * x^k))) \\ Indranil Ghosh, Apr 03 2017

Formula

G.f.: 1/(1 - Sum_{p prime, k>=2} x^(p^k)).
Showing 1-7 of 7 results.

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